quadp (p42, positive)

Percentage Accurate: 51.4% → 87.0%

Time: 7.3s

Alternatives: 8

Speedup: 2.5×

Specification

Math

\[\begin{array}{l} \\ \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (+ (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)))

C

double code(double a, double b, double c) {
	return (-b + sqrt(((b * b) - (4.0 * (a * c))))) / (2.0 * a);
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = (-b + sqrt(((b * b) - (4.0d0 * (a * c))))) / (2.0d0 * a)
end function

Java

public static double code(double a, double b, double c) {
	return (-b + Math.sqrt(((b * b) - (4.0 * (a * c))))) / (2.0 * a);
}

Python

def code(a, b, c):
	return (-b + math.sqrt(((b * b) - (4.0 * (a * c))))) / (2.0 * a)

Julia

function code(a, b, c)
	return Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(4.0 * Float64(a * c))))) / Float64(2.0 * a))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (-b + sqrt(((b * b) - (4.0 * (a * c))))) / (2.0 * a);
end

Wolfram

code[a_, b_, c_] := N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(4.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]

Initial Program: 51.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (+ (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)))

C

double code(double a, double b, double c) {
	return (-b + sqrt(((b * b) - (4.0 * (a * c))))) / (2.0 * a);
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = (-b + sqrt(((b * b) - (4.0d0 * (a * c))))) / (2.0d0 * a)
end function

Java

public static double code(double a, double b, double c) {
	return (-b + Math.sqrt(((b * b) - (4.0 * (a * c))))) / (2.0 * a);
}

Python

def code(a, b, c):
	return (-b + math.sqrt(((b * b) - (4.0 * (a * c))))) / (2.0 * a)

Julia

function code(a, b, c)
	return Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(4.0 * Float64(a * c))))) / Float64(2.0 * a))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (-b + sqrt(((b * b) - (4.0 * (a * c))))) / (2.0 * a);
end

Wolfram

code[a_, b_, c_] := N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(4.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]

Alternative 1: 87.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -6 \cdot 10^{+104}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 9 \cdot 10^{-195}:\\ \;\;\;\;\frac{b - \sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)}}{-2 \cdot a}\\ \mathbf{elif}\;b \leq 2 \cdot 10^{-31}:\\ \;\;\;\;\frac{\frac{\left(c \cdot a\right) \cdot -4}{\sqrt{\mathsf{fma}\left(c \cdot -4, a, b \cdot b\right)} + b}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -6e+104)
   (- (/ c b) (/ b a))
   (if (<= b 9e-195)
     (/ (- b (sqrt (fma -4.0 (* c a) (* b b)))) (* -2.0 a))
     (if (<= b 2e-31)
       (/
        (/ (* (* c a) -4.0) (+ (sqrt (fma (* c -4.0) a (* b b))) b))
        (* 2.0 a))
       (/ (- c) b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -6e+104) {
		tmp = (c / b) - (b / a);
	} else if (b <= 9e-195) {
		tmp = (b - sqrt(fma(-4.0, (c * a), (b * b)))) / (-2.0 * a);
	} else if (b <= 2e-31) {
		tmp = (((c * a) * -4.0) / (sqrt(fma((c * -4.0), a, (b * b))) + b)) / (2.0 * a);
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -6e+104)
		tmp = Float64(Float64(c / b) - Float64(b / a));
	elseif (b <= 9e-195)
		tmp = Float64(Float64(b - sqrt(fma(-4.0, Float64(c * a), Float64(b * b)))) / Float64(-2.0 * a));
	elseif (b <= 2e-31)
		tmp = Float64(Float64(Float64(Float64(c * a) * -4.0) / Float64(sqrt(fma(Float64(c * -4.0), a, Float64(b * b))) + b)) / Float64(2.0 * a));
	else
		tmp = Float64(Float64(-c) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -6e+104], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 9e-195], N[(N[(b - N[Sqrt[N[(-4.0 * N[(c * a), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(-2.0 * a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 2e-31], N[(N[(N[(N[(c * a), $MachinePrecision] * -4.0), $MachinePrecision] / N[(N[Sqrt[N[(N[(c * -4.0), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[((-c) / b), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if b < -5.99999999999999937e104

   1. Initial program 59.1%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. Add Preprocessing

   3. Taylor expanded in b around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(b \cdot \left(-1 \cdot \frac{c}{{b}^{2}} + \frac{1}{a}\right)\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(b \cdot \left(-1 \cdot \frac{c}{{b}^{2}} + \frac{1}{a}\right)\right)} \]

      2. lower-neg.f64 N/A

         \[\leadsto \color{blue}{-b \cdot \left(-1 \cdot \frac{c}{{b}^{2}} + \frac{1}{a}\right)} \]

      3. distribute-rgt-in N/A

         \[\leadsto -\color{blue}{\left(\left(-1 \cdot \frac{c}{{b}^{2}}\right) \cdot b + \frac{1}{a} \cdot b\right)} \]

      4. associate-*l/ N/A

         \[\leadsto -\left(\left(-1 \cdot \frac{c}{{b}^{2}}\right) \cdot b + \color{blue}{\frac{1 \cdot b}{a}}\right) \]

      5. *-lft-identity N/A

         \[\leadsto -\left(\left(-1 \cdot \frac{c}{{b}^{2}}\right) \cdot b + \frac{\color{blue}{b}}{a}\right) \]

      6. lower-fma.f64 N/A

         \[\leadsto -\color{blue}{\mathsf{fma}\left(-1 \cdot \frac{c}{{b}^{2}}, b, \frac{b}{a}\right)} \]

      7. associate-*r/ N/A

         \[\leadsto -\mathsf{fma}\left(\color{blue}{\frac{-1 \cdot c}{{b}^{2}}}, b, \frac{b}{a}\right) \]

      8. mul-1-neg N/A

         \[\leadsto -\mathsf{fma}\left(\frac{\color{blue}{\mathsf{neg}\left(c\right)}}{{b}^{2}}, b, \frac{b}{a}\right) \]

      9. lower-/.f64 N/A

         \[\leadsto -\mathsf{fma}\left(\color{blue}{\frac{\mathsf{neg}\left(c\right)}{{b}^{2}}}, b, \frac{b}{a}\right) \]

      10. lower-neg.f64 N/A

          \[\leadsto -\mathsf{fma}\left(\frac{\color{blue}{-c}}{{b}^{2}}, b, \frac{b}{a}\right) \]

      11. unpow2 N/A

          \[\leadsto -\mathsf{fma}\left(\frac{-c}{\color{blue}{b \cdot b}}, b, \frac{b}{a}\right) \]

      12. lower-*.f64 N/A

          \[\leadsto -\mathsf{fma}\left(\frac{-c}{\color{blue}{b \cdot b}}, b, \frac{b}{a}\right) \]

      13. lower-/.f64 95.6

          \[\leadsto -\mathsf{fma}\left(\frac{-c}{b \cdot b}, b, \color{blue}{\frac{b}{a}}\right) \]

   5. Applied rewrites 95.6%

      \[\leadsto \color{blue}{-\mathsf{fma}\left(\frac{-c}{b \cdot b}, b, \frac{b}{a}\right)} \]

   6. Taylor expanded in c around 0

      \[\leadsto \frac{c}{b} - \color{blue}{\frac{b}{a}} \]
   7. Step-by-step derivation

   8. Applied rewrites 95.6%

      \[\leadsto \frac{c}{b} - \color{blue}{\frac{b}{a}} \]

   if -5.99999999999999937e104 < b < 9e-195

   1. Initial program 82.1%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. Add Preprocessing

   4. Applied rewrites 82.1%

      \[\leadsto \color{blue}{\frac{b - \sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)}}{-2 \cdot a}} \]

   if 9e-195 < b < 2e-31

   1. Initial program 52.2%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. Add Preprocessing

   4. Applied rewrites 52.7%

      \[\leadsto \frac{\color{blue}{\frac{\mathsf{fma}\left(-b, b, \mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)\right)}{\sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)} + b}}}{2 \cdot a} \]

   5. Taylor expanded in a around 0

      \[\leadsto \frac{\frac{\color{blue}{-4 \cdot \left(a \cdot c\right) + \left(-1 \cdot {b}^{2} + {b}^{2}\right)}}{\sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)} + b}}{2 \cdot a} \]
   6. Step-by-step derivation

      1. distribute-lft1-in N/A

         \[\leadsto \frac{\frac{-4 \cdot \left(a \cdot c\right) + \color{blue}{\left(-1 + 1\right) \cdot {b}^{2}}}{\sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)} + b}}{2 \cdot a} \]

      2. metadata-eval N/A

         \[\leadsto \frac{\frac{-4 \cdot \left(a \cdot c\right) + \color{blue}{0} \cdot {b}^{2}}{\sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)} + b}}{2 \cdot a} \]

      3. mul0-lft N/A

         \[\leadsto \frac{\frac{-4 \cdot \left(a \cdot c\right) + \color{blue}{0}}{\sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)} + b}}{2 \cdot a} \]

      4. mul0-lft N/A

         \[\leadsto \frac{\frac{-4 \cdot \left(a \cdot c\right) + \color{blue}{0 \cdot \frac{{b}^{2}}{c}}}{\sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)} + b}}{2 \cdot a} \]

      5. metadata-eval N/A

         \[\leadsto \frac{\frac{-4 \cdot \left(a \cdot c\right) + \color{blue}{\left(-1 + 1\right)} \cdot \frac{{b}^{2}}{c}}{\sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)} + b}}{2 \cdot a} \]

      6. distribute-lft1-in N/A

         \[\leadsto \frac{\frac{-4 \cdot \left(a \cdot c\right) + \color{blue}{\left(-1 \cdot \frac{{b}^{2}}{c} + \frac{{b}^{2}}{c}\right)}}{\sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)} + b}}{2 \cdot a} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{\mathsf{fma}\left(-4, a \cdot c, -1 \cdot \frac{{b}^{2}}{c} + \frac{{b}^{2}}{c}\right)}}{\sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)} + b}}{2 \cdot a} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(-4, \color{blue}{a \cdot c}, -1 \cdot \frac{{b}^{2}}{c} + \frac{{b}^{2}}{c}\right)}{\sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)} + b}}{2 \cdot a} \]

      9. distribute-lft1-in N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(-4, a \cdot c, \color{blue}{\left(-1 + 1\right) \cdot \frac{{b}^{2}}{c}}\right)}{\sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)} + b}}{2 \cdot a} \]

      10. metadata-eval N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(-4, a \cdot c, \color{blue}{0} \cdot \frac{{b}^{2}}{c}\right)}{\sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)} + b}}{2 \cdot a} \]

      11. mul0-lft 73.5

          \[\leadsto \frac{\frac{\mathsf{fma}\left(-4, a \cdot c, \color{blue}{0}\right)}{\sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)} + b}}{2 \cdot a} \]

   7. Applied rewrites 73.5%

      \[\leadsto \frac{\frac{\color{blue}{\mathsf{fma}\left(-4, a \cdot c, 0\right)}}{\sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)} + b}}{2 \cdot a} \]
   8. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \frac{\frac{\mathsf{Rewrite<=}\left(lift-*.f64, \left(c \cdot a\right)\right) \cdot -4}{\sqrt{\color{blue}{-4 \cdot \left(c \cdot a\right) + b \cdot b}} + b}}{2 \cdot a} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\frac{\mathsf{Rewrite<=}\left(lift-*.f64, \left(c \cdot a\right)\right) \cdot -4}{\sqrt{-4 \cdot \color{blue}{\left(c \cdot a\right)} + b \cdot b} + b}}{2 \cdot a} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\frac{\mathsf{Rewrite<=}\left(lift-*.f64, \left(c \cdot a\right)\right) \cdot -4}{\sqrt{\color{blue}{\left(-4 \cdot c\right) \cdot a} + b \cdot b} + b}}{2 \cdot a} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\frac{\mathsf{Rewrite<=}\left(lift-*.f64, \left(c \cdot a\right)\right) \cdot -4}{\sqrt{\color{blue}{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)}} + b}}{2 \cdot a} \]

      5. *-commutative N/A

         \[\leadsto \frac{\frac{\mathsf{Rewrite<=}\left(lift-*.f64, \left(c \cdot a\right)\right) \cdot -4}{\sqrt{\mathsf{fma}\left(\color{blue}{c \cdot -4}, a, b \cdot b\right)} + b}}{2 \cdot a} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\frac{\mathsf{Rewrite<=}\left(lift-*.f64, \left(c \cdot a\right)\right) \cdot -4}{\sqrt{\mathsf{fma}\left(\color{blue}{c \cdot -4}, a, b \cdot b\right)} + b}}{2 \cdot a} \]

   9. Applied rewrites 73.5%

      \[\leadsto \color{blue}{\frac{\frac{\left(c \cdot a\right) \cdot -4}{\sqrt{\mathsf{fma}\left(c \cdot -4, a, b \cdot b\right)} + b}}{2 \cdot a}} \]

   if 2e-31 < b

   1. Initial program 12.0%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{-1 \cdot c}{b}} \]

      2. mul-1-neg N/A

         \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(c\right)}}{b} \]

      3. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(c\right)}{b}} \]

      4. lower-neg.f64 94.8

         \[\leadsto \frac{\color{blue}{-c}}{b} \]

   5. Applied rewrites 94.8%

      \[\leadsto \color{blue}{\frac{-c}{b}} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 2: 86.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -6 \cdot 10^{+104}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 6.3 \cdot 10^{-104}:\\ \;\;\;\;\frac{b - \sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)}}{-2 \cdot a}\\ \mathbf{elif}\;b \leq 1.45 \cdot 10^{-31}:\\ \;\;\;\;\frac{\left(c \cdot a\right) \cdot -4}{\left(2 \cdot a\right) \cdot \left(\sqrt{\mathsf{fma}\left(c \cdot -4, a, b \cdot b\right)} + b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -6e+104)
   (- (/ c b) (/ b a))
   (if (<= b 6.3e-104)
     (/ (- b (sqrt (fma -4.0 (* c a) (* b b)))) (* -2.0 a))
     (if (<= b 1.45e-31)
       (/
        (* (* c a) -4.0)
        (* (* 2.0 a) (+ (sqrt (fma (* c -4.0) a (* b b))) b)))
       (/ (- c) b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -6e+104) {
		tmp = (c / b) - (b / a);
	} else if (b <= 6.3e-104) {
		tmp = (b - sqrt(fma(-4.0, (c * a), (b * b)))) / (-2.0 * a);
	} else if (b <= 1.45e-31) {
		tmp = ((c * a) * -4.0) / ((2.0 * a) * (sqrt(fma((c * -4.0), a, (b * b))) + b));
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -6e+104)
		tmp = Float64(Float64(c / b) - Float64(b / a));
	elseif (b <= 6.3e-104)
		tmp = Float64(Float64(b - sqrt(fma(-4.0, Float64(c * a), Float64(b * b)))) / Float64(-2.0 * a));
	elseif (b <= 1.45e-31)
		tmp = Float64(Float64(Float64(c * a) * -4.0) / Float64(Float64(2.0 * a) * Float64(sqrt(fma(Float64(c * -4.0), a, Float64(b * b))) + b)));
	else
		tmp = Float64(Float64(-c) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -6e+104], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 6.3e-104], N[(N[(b - N[Sqrt[N[(-4.0 * N[(c * a), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(-2.0 * a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.45e-31], N[(N[(N[(c * a), $MachinePrecision] * -4.0), $MachinePrecision] / N[(N[(2.0 * a), $MachinePrecision] * N[(N[Sqrt[N[(N[(c * -4.0), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-c) / b), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if b < -5.99999999999999937e104

   1. Initial program 59.1%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. Add Preprocessing

   3. Taylor expanded in b around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(b \cdot \left(-1 \cdot \frac{c}{{b}^{2}} + \frac{1}{a}\right)\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(b \cdot \left(-1 \cdot \frac{c}{{b}^{2}} + \frac{1}{a}\right)\right)} \]

      2. lower-neg.f64 N/A

         \[\leadsto \color{blue}{-b \cdot \left(-1 \cdot \frac{c}{{b}^{2}} + \frac{1}{a}\right)} \]

      3. distribute-rgt-in N/A

         \[\leadsto -\color{blue}{\left(\left(-1 \cdot \frac{c}{{b}^{2}}\right) \cdot b + \frac{1}{a} \cdot b\right)} \]

      4. associate-*l/ N/A

         \[\leadsto -\left(\left(-1 \cdot \frac{c}{{b}^{2}}\right) \cdot b + \color{blue}{\frac{1 \cdot b}{a}}\right) \]

      5. *-lft-identity N/A

         \[\leadsto -\left(\left(-1 \cdot \frac{c}{{b}^{2}}\right) \cdot b + \frac{\color{blue}{b}}{a}\right) \]

      6. lower-fma.f64 N/A

         \[\leadsto -\color{blue}{\mathsf{fma}\left(-1 \cdot \frac{c}{{b}^{2}}, b, \frac{b}{a}\right)} \]

      7. associate-*r/ N/A

         \[\leadsto -\mathsf{fma}\left(\color{blue}{\frac{-1 \cdot c}{{b}^{2}}}, b, \frac{b}{a}\right) \]

      8. mul-1-neg N/A

         \[\leadsto -\mathsf{fma}\left(\frac{\color{blue}{\mathsf{neg}\left(c\right)}}{{b}^{2}}, b, \frac{b}{a}\right) \]

      9. lower-/.f64 N/A

         \[\leadsto -\mathsf{fma}\left(\color{blue}{\frac{\mathsf{neg}\left(c\right)}{{b}^{2}}}, b, \frac{b}{a}\right) \]

      10. lower-neg.f64 N/A

          \[\leadsto -\mathsf{fma}\left(\frac{\color{blue}{-c}}{{b}^{2}}, b, \frac{b}{a}\right) \]

      11. unpow2 N/A

          \[\leadsto -\mathsf{fma}\left(\frac{-c}{\color{blue}{b \cdot b}}, b, \frac{b}{a}\right) \]

      12. lower-*.f64 N/A

          \[\leadsto -\mathsf{fma}\left(\frac{-c}{\color{blue}{b \cdot b}}, b, \frac{b}{a}\right) \]

      13. lower-/.f64 95.6

          \[\leadsto -\mathsf{fma}\left(\frac{-c}{b \cdot b}, b, \color{blue}{\frac{b}{a}}\right) \]

   5. Applied rewrites 95.6%

      \[\leadsto \color{blue}{-\mathsf{fma}\left(\frac{-c}{b \cdot b}, b, \frac{b}{a}\right)} \]

   6. Taylor expanded in c around 0

      \[\leadsto \frac{c}{b} - \color{blue}{\frac{b}{a}} \]
   7. Step-by-step derivation

   8. Applied rewrites 95.6%

      \[\leadsto \frac{c}{b} - \color{blue}{\frac{b}{a}} \]

   if -5.99999999999999937e104 < b < 6.29999999999999965e-104

   1. Initial program 76.4%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. Add Preprocessing

   4. Applied rewrites 76.4%

      \[\leadsto \color{blue}{\frac{b - \sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)}}{-2 \cdot a}} \]

   if 6.29999999999999965e-104 < b < 1.45e-31

   1. Initial program 57.4%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. Add Preprocessing

   4. Applied rewrites 58.3%

      \[\leadsto \frac{\color{blue}{\frac{\mathsf{fma}\left(-b, b, \mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)\right)}{\sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)} + b}}}{2 \cdot a} \]

   5. Taylor expanded in a around 0

      \[\leadsto \frac{\frac{\color{blue}{-4 \cdot \left(a \cdot c\right) + \left(-1 \cdot {b}^{2} + {b}^{2}\right)}}{\sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)} + b}}{2 \cdot a} \]
   6. Step-by-step derivation

      1. distribute-lft1-in N/A

         \[\leadsto \frac{\frac{-4 \cdot \left(a \cdot c\right) + \color{blue}{\left(-1 + 1\right) \cdot {b}^{2}}}{\sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)} + b}}{2 \cdot a} \]

      2. metadata-eval N/A

         \[\leadsto \frac{\frac{-4 \cdot \left(a \cdot c\right) + \color{blue}{0} \cdot {b}^{2}}{\sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)} + b}}{2 \cdot a} \]

      3. mul0-lft N/A

         \[\leadsto \frac{\frac{-4 \cdot \left(a \cdot c\right) + \color{blue}{0}}{\sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)} + b}}{2 \cdot a} \]

      4. mul0-lft N/A

         \[\leadsto \frac{\frac{-4 \cdot \left(a \cdot c\right) + \color{blue}{0 \cdot \frac{{b}^{2}}{c}}}{\sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)} + b}}{2 \cdot a} \]

      5. metadata-eval N/A

         \[\leadsto \frac{\frac{-4 \cdot \left(a \cdot c\right) + \color{blue}{\left(-1 + 1\right)} \cdot \frac{{b}^{2}}{c}}{\sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)} + b}}{2 \cdot a} \]

      6. distribute-lft1-in N/A

         \[\leadsto \frac{\frac{-4 \cdot \left(a \cdot c\right) + \color{blue}{\left(-1 \cdot \frac{{b}^{2}}{c} + \frac{{b}^{2}}{c}\right)}}{\sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)} + b}}{2 \cdot a} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{\mathsf{fma}\left(-4, a \cdot c, -1 \cdot \frac{{b}^{2}}{c} + \frac{{b}^{2}}{c}\right)}}{\sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)} + b}}{2 \cdot a} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(-4, \color{blue}{a \cdot c}, -1 \cdot \frac{{b}^{2}}{c} + \frac{{b}^{2}}{c}\right)}{\sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)} + b}}{2 \cdot a} \]

      9. distribute-lft1-in N/A

         \[\leadsto \frac{\frac{\mathsf{fma}\left(-4, a \cdot c, \color{blue}{\left(-1 + 1\right) \cdot \frac{{b}^{2}}{c}}\right)}{\sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)} + b}}{2 \cdot a} \]

      10. metadata-eval N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(-4, a \cdot c, \color{blue}{0} \cdot \frac{{b}^{2}}{c}\right)}{\sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)} + b}}{2 \cdot a} \]

      11. mul0-lft 90.3

          \[\leadsto \frac{\frac{\mathsf{fma}\left(-4, a \cdot c, \color{blue}{0}\right)}{\sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)} + b}}{2 \cdot a} \]

   7. Applied rewrites 90.3%

      \[\leadsto \frac{\frac{\color{blue}{\mathsf{fma}\left(-4, a \cdot c, 0\right)}}{\sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)} + b}}{2 \cdot a} \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(-4, a \cdot c, 0\right)}{\sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)} + b}}{2 \cdot a}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{\mathsf{fma}\left(-4, a \cdot c, 0\right)}{\sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)} + b}}}{2 \cdot a} \]

      3. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-4, a \cdot c, 0\right)}{\left(\sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)} + b\right) \cdot \left(2 \cdot a\right)}} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-4, a \cdot c, 0\right)}{\left(\sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)} + b\right) \cdot \left(2 \cdot a\right)}} \]

      5. *-commutative N/A

         \[\leadsto \frac{\mathsf{Rewrite<=}\left(lift-*.f64, \left(c \cdot a\right)\right) \cdot -4}{\color{blue}{\left(2 \cdot a\right) \cdot \left(\sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)} + b\right)}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{Rewrite<=}\left(lift-*.f64, \left(c \cdot a\right)\right) \cdot -4}{\color{blue}{\left(2 \cdot a\right) \cdot \left(\sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)} + b\right)}} \]

   9. Applied rewrites 85.6%

      \[\leadsto \color{blue}{\frac{\left(c \cdot a\right) \cdot -4}{\left(2 \cdot a\right) \cdot \left(\sqrt{\mathsf{fma}\left(c \cdot -4, a, b \cdot b\right)} + b\right)}} \]

   if 1.45e-31 < b

   1. Initial program 12.0%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{-1 \cdot c}{b}} \]

      2. mul-1-neg N/A

         \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(c\right)}}{b} \]

      3. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(c\right)}{b}} \]

      4. lower-neg.f64 94.8

         \[\leadsto \frac{\color{blue}{-c}}{b} \]

   5. Applied rewrites 94.8%

      \[\leadsto \color{blue}{\frac{-c}{b}} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 3: 85.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -6 \cdot 10^{+104}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 1.3 \cdot 10^{-92}:\\ \;\;\;\;\frac{b - \sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)}}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -6e+104)
   (- (/ c b) (/ b a))
   (if (<= b 1.3e-92)
     (/ (- b (sqrt (fma -4.0 (* c a) (* b b)))) (* -2.0 a))
     (/ (- c) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -6e+104) {
		tmp = (c / b) - (b / a);
	} else if (b <= 1.3e-92) {
		tmp = (b - sqrt(fma(-4.0, (c * a), (b * b)))) / (-2.0 * a);
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -6e+104)
		tmp = Float64(Float64(c / b) - Float64(b / a));
	elseif (b <= 1.3e-92)
		tmp = Float64(Float64(b - sqrt(fma(-4.0, Float64(c * a), Float64(b * b)))) / Float64(-2.0 * a));
	else
		tmp = Float64(Float64(-c) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -6e+104], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.3e-92], N[(N[(b - N[Sqrt[N[(-4.0 * N[(c * a), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(-2.0 * a), $MachinePrecision]), $MachinePrecision], N[((-c) / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -5.99999999999999937e104

   1. Initial program 59.1%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. Add Preprocessing

   3. Taylor expanded in b around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(b \cdot \left(-1 \cdot \frac{c}{{b}^{2}} + \frac{1}{a}\right)\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(b \cdot \left(-1 \cdot \frac{c}{{b}^{2}} + \frac{1}{a}\right)\right)} \]

      2. lower-neg.f64 N/A

         \[\leadsto \color{blue}{-b \cdot \left(-1 \cdot \frac{c}{{b}^{2}} + \frac{1}{a}\right)} \]

      3. distribute-rgt-in N/A

         \[\leadsto -\color{blue}{\left(\left(-1 \cdot \frac{c}{{b}^{2}}\right) \cdot b + \frac{1}{a} \cdot b\right)} \]

      4. associate-*l/ N/A

         \[\leadsto -\left(\left(-1 \cdot \frac{c}{{b}^{2}}\right) \cdot b + \color{blue}{\frac{1 \cdot b}{a}}\right) \]

      5. *-lft-identity N/A

         \[\leadsto -\left(\left(-1 \cdot \frac{c}{{b}^{2}}\right) \cdot b + \frac{\color{blue}{b}}{a}\right) \]

      6. lower-fma.f64 N/A

         \[\leadsto -\color{blue}{\mathsf{fma}\left(-1 \cdot \frac{c}{{b}^{2}}, b, \frac{b}{a}\right)} \]

      7. associate-*r/ N/A

         \[\leadsto -\mathsf{fma}\left(\color{blue}{\frac{-1 \cdot c}{{b}^{2}}}, b, \frac{b}{a}\right) \]

      8. mul-1-neg N/A

         \[\leadsto -\mathsf{fma}\left(\frac{\color{blue}{\mathsf{neg}\left(c\right)}}{{b}^{2}}, b, \frac{b}{a}\right) \]

      9. lower-/.f64 N/A

         \[\leadsto -\mathsf{fma}\left(\color{blue}{\frac{\mathsf{neg}\left(c\right)}{{b}^{2}}}, b, \frac{b}{a}\right) \]

      10. lower-neg.f64 N/A

          \[\leadsto -\mathsf{fma}\left(\frac{\color{blue}{-c}}{{b}^{2}}, b, \frac{b}{a}\right) \]

      11. unpow2 N/A

          \[\leadsto -\mathsf{fma}\left(\frac{-c}{\color{blue}{b \cdot b}}, b, \frac{b}{a}\right) \]

      12. lower-*.f64 N/A

          \[\leadsto -\mathsf{fma}\left(\frac{-c}{\color{blue}{b \cdot b}}, b, \frac{b}{a}\right) \]

      13. lower-/.f64 95.6

          \[\leadsto -\mathsf{fma}\left(\frac{-c}{b \cdot b}, b, \color{blue}{\frac{b}{a}}\right) \]

   5. Applied rewrites 95.6%

      \[\leadsto \color{blue}{-\mathsf{fma}\left(\frac{-c}{b \cdot b}, b, \frac{b}{a}\right)} \]

   6. Taylor expanded in c around 0

      \[\leadsto \frac{c}{b} - \color{blue}{\frac{b}{a}} \]
   7. Step-by-step derivation

   8. Applied rewrites 95.6%

      \[\leadsto \frac{c}{b} - \color{blue}{\frac{b}{a}} \]

   if -5.99999999999999937e104 < b < 1.3e-92

   1. Initial program 77.3%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. Add Preprocessing

   4. Applied rewrites 77.3%

      \[\leadsto \color{blue}{\frac{b - \sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)}}{-2 \cdot a}} \]

   if 1.3e-92 < b

   1. Initial program 17.7%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{-1 \cdot c}{b}} \]

      2. mul-1-neg N/A

         \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(c\right)}}{b} \]

      3. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(c\right)}{b}} \]

      4. lower-neg.f64 89.1

         \[\leadsto \frac{\color{blue}{-c}}{b} \]

   5. Applied rewrites 89.1%

      \[\leadsto \color{blue}{\frac{-c}{b}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 4: 81.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.4 \cdot 10^{-83}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 1.3 \cdot 10^{-92}:\\ \;\;\;\;\frac{b - \sqrt{-4 \cdot \left(a \cdot c\right)}}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.4e-83)
   (- (/ c b) (/ b a))
   (if (<= b 1.3e-92)
     (/ (- b (sqrt (* -4.0 (* a c)))) (* -2.0 a))
     (/ (- c) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.4e-83) {
		tmp = (c / b) - (b / a);
	} else if (b <= 1.3e-92) {
		tmp = (b - sqrt((-4.0 * (a * c)))) / (-2.0 * a);
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-1.4d-83)) then
        tmp = (c / b) - (b / a)
    else if (b <= 1.3d-92) then
        tmp = (b - sqrt(((-4.0d0) * (a * c)))) / ((-2.0d0) * a)
    else
        tmp = -c / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.4e-83) {
		tmp = (c / b) - (b / a);
	} else if (b <= 1.3e-92) {
		tmp = (b - Math.sqrt((-4.0 * (a * c)))) / (-2.0 * a);
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1.4e-83:
		tmp = (c / b) - (b / a)
	elif b <= 1.3e-92:
		tmp = (b - math.sqrt((-4.0 * (a * c)))) / (-2.0 * a)
	else:
		tmp = -c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.4e-83)
		tmp = Float64(Float64(c / b) - Float64(b / a));
	elseif (b <= 1.3e-92)
		tmp = Float64(Float64(b - sqrt(Float64(-4.0 * Float64(a * c)))) / Float64(-2.0 * a));
	else
		tmp = Float64(Float64(-c) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -1.4e-83)
		tmp = (c / b) - (b / a);
	elseif (b <= 1.3e-92)
		tmp = (b - sqrt((-4.0 * (a * c)))) / (-2.0 * a);
	else
		tmp = -c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.4e-83], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.3e-92], N[(N[(b - N[Sqrt[N[(-4.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(-2.0 * a), $MachinePrecision]), $MachinePrecision], N[((-c) / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.4e-83

   1. Initial program 72.3%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. Add Preprocessing

   3. Taylor expanded in b around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(b \cdot \left(-1 \cdot \frac{c}{{b}^{2}} + \frac{1}{a}\right)\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(b \cdot \left(-1 \cdot \frac{c}{{b}^{2}} + \frac{1}{a}\right)\right)} \]

      2. lower-neg.f64 N/A

         \[\leadsto \color{blue}{-b \cdot \left(-1 \cdot \frac{c}{{b}^{2}} + \frac{1}{a}\right)} \]

      3. distribute-rgt-in N/A

         \[\leadsto -\color{blue}{\left(\left(-1 \cdot \frac{c}{{b}^{2}}\right) \cdot b + \frac{1}{a} \cdot b\right)} \]

      4. associate-*l/ N/A

         \[\leadsto -\left(\left(-1 \cdot \frac{c}{{b}^{2}}\right) \cdot b + \color{blue}{\frac{1 \cdot b}{a}}\right) \]

      5. *-lft-identity N/A

         \[\leadsto -\left(\left(-1 \cdot \frac{c}{{b}^{2}}\right) \cdot b + \frac{\color{blue}{b}}{a}\right) \]

      6. lower-fma.f64 N/A

         \[\leadsto -\color{blue}{\mathsf{fma}\left(-1 \cdot \frac{c}{{b}^{2}}, b, \frac{b}{a}\right)} \]

      7. associate-*r/ N/A

         \[\leadsto -\mathsf{fma}\left(\color{blue}{\frac{-1 \cdot c}{{b}^{2}}}, b, \frac{b}{a}\right) \]

      8. mul-1-neg N/A

         \[\leadsto -\mathsf{fma}\left(\frac{\color{blue}{\mathsf{neg}\left(c\right)}}{{b}^{2}}, b, \frac{b}{a}\right) \]

      9. lower-/.f64 N/A

         \[\leadsto -\mathsf{fma}\left(\color{blue}{\frac{\mathsf{neg}\left(c\right)}{{b}^{2}}}, b, \frac{b}{a}\right) \]

      10. lower-neg.f64 N/A

          \[\leadsto -\mathsf{fma}\left(\frac{\color{blue}{-c}}{{b}^{2}}, b, \frac{b}{a}\right) \]

      11. unpow2 N/A

          \[\leadsto -\mathsf{fma}\left(\frac{-c}{\color{blue}{b \cdot b}}, b, \frac{b}{a}\right) \]

      12. lower-*.f64 N/A

          \[\leadsto -\mathsf{fma}\left(\frac{-c}{\color{blue}{b \cdot b}}, b, \frac{b}{a}\right) \]

      13. lower-/.f64 88.7

          \[\leadsto -\mathsf{fma}\left(\frac{-c}{b \cdot b}, b, \color{blue}{\frac{b}{a}}\right) \]

   5. Applied rewrites 88.7%

      \[\leadsto \color{blue}{-\mathsf{fma}\left(\frac{-c}{b \cdot b}, b, \frac{b}{a}\right)} \]

   6. Taylor expanded in c around 0

      \[\leadsto \frac{c}{b} - \color{blue}{\frac{b}{a}} \]
   7. Step-by-step derivation

   8. Applied rewrites 88.7%

      \[\leadsto \frac{c}{b} - \color{blue}{\frac{b}{a}} \]

   if -1.4e-83 < b < 1.3e-92

   1. Initial program 72.0%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. Add Preprocessing

   4. Applied rewrites 72.0%

      \[\leadsto \color{blue}{\frac{b - \sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)}}{-2 \cdot a}} \]

   5. Taylor expanded in a around inf

      \[\leadsto \frac{b - \sqrt{\color{blue}{-4 \cdot \left(a \cdot c\right)}}}{-2 \cdot a} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{b - \sqrt{\color{blue}{-4 \cdot \left(a \cdot c\right)}}}{-2 \cdot a} \]

      2. lower-*.f64 66.4

         \[\leadsto \frac{b - \sqrt{-4 \cdot \color{blue}{\left(a \cdot c\right)}}}{-2 \cdot a} \]

   7. Applied rewrites 66.4%

      \[\leadsto \frac{b - \sqrt{\color{blue}{-4 \cdot \left(a \cdot c\right)}}}{-2 \cdot a} \]

   if 1.3e-92 < b

   1. Initial program 17.7%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{-1 \cdot c}{b}} \]

      2. mul-1-neg N/A

         \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(c\right)}}{b} \]

      3. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(c\right)}{b}} \]

      4. lower-neg.f64 89.1

         \[\leadsto \frac{\color{blue}{-c}}{b} \]

   5. Applied rewrites 89.1%

      \[\leadsto \color{blue}{\frac{-c}{b}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 5: 68.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -5e-310) (- (/ c b) (/ b a)) (/ (- c) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e-310) {
		tmp = (c / b) - (b / a);
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-5d-310)) then
        tmp = (c / b) - (b / a)
    else
        tmp = -c / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e-310) {
		tmp = (c / b) - (b / a);
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -5e-310:
		tmp = (c / b) - (b / a)
	else:
		tmp = -c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -5e-310)
		tmp = Float64(Float64(c / b) - Float64(b / a));
	else
		tmp = Float64(Float64(-c) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -5e-310)
		tmp = (c / b) - (b / a);
	else
		tmp = -c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -5e-310], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], N[((-c) / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -4.999999999999985e-310

   1. Initial program 74.0%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. Add Preprocessing

   3. Taylor expanded in b around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(b \cdot \left(-1 \cdot \frac{c}{{b}^{2}} + \frac{1}{a}\right)\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(b \cdot \left(-1 \cdot \frac{c}{{b}^{2}} + \frac{1}{a}\right)\right)} \]

      2. lower-neg.f64 N/A

         \[\leadsto \color{blue}{-b \cdot \left(-1 \cdot \frac{c}{{b}^{2}} + \frac{1}{a}\right)} \]

      3. distribute-rgt-in N/A

         \[\leadsto -\color{blue}{\left(\left(-1 \cdot \frac{c}{{b}^{2}}\right) \cdot b + \frac{1}{a} \cdot b\right)} \]

      4. associate-*l/ N/A

         \[\leadsto -\left(\left(-1 \cdot \frac{c}{{b}^{2}}\right) \cdot b + \color{blue}{\frac{1 \cdot b}{a}}\right) \]

      5. *-lft-identity N/A

         \[\leadsto -\left(\left(-1 \cdot \frac{c}{{b}^{2}}\right) \cdot b + \frac{\color{blue}{b}}{a}\right) \]

      6. lower-fma.f64 N/A

         \[\leadsto -\color{blue}{\mathsf{fma}\left(-1 \cdot \frac{c}{{b}^{2}}, b, \frac{b}{a}\right)} \]

      7. associate-*r/ N/A

         \[\leadsto -\mathsf{fma}\left(\color{blue}{\frac{-1 \cdot c}{{b}^{2}}}, b, \frac{b}{a}\right) \]

      8. mul-1-neg N/A

         \[\leadsto -\mathsf{fma}\left(\frac{\color{blue}{\mathsf{neg}\left(c\right)}}{{b}^{2}}, b, \frac{b}{a}\right) \]

      9. lower-/.f64 N/A

         \[\leadsto -\mathsf{fma}\left(\color{blue}{\frac{\mathsf{neg}\left(c\right)}{{b}^{2}}}, b, \frac{b}{a}\right) \]

      10. lower-neg.f64 N/A

          \[\leadsto -\mathsf{fma}\left(\frac{\color{blue}{-c}}{{b}^{2}}, b, \frac{b}{a}\right) \]

      11. unpow2 N/A

          \[\leadsto -\mathsf{fma}\left(\frac{-c}{\color{blue}{b \cdot b}}, b, \frac{b}{a}\right) \]

      12. lower-*.f64 N/A

          \[\leadsto -\mathsf{fma}\left(\frac{-c}{\color{blue}{b \cdot b}}, b, \frac{b}{a}\right) \]

      13. lower-/.f64 70.8

          \[\leadsto -\mathsf{fma}\left(\frac{-c}{b \cdot b}, b, \color{blue}{\frac{b}{a}}\right) \]

   5. Applied rewrites 70.8%

      \[\leadsto \color{blue}{-\mathsf{fma}\left(\frac{-c}{b \cdot b}, b, \frac{b}{a}\right)} \]

   6. Taylor expanded in c around 0

      \[\leadsto \frac{c}{b} - \color{blue}{\frac{b}{a}} \]
   7. Step-by-step derivation

   8. Applied rewrites 71.0%

      \[\leadsto \frac{c}{b} - \color{blue}{\frac{b}{a}} \]

   if -4.999999999999985e-310 < b

   1. Initial program 30.5%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{-1 \cdot c}{b}} \]

      2. mul-1-neg N/A

         \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(c\right)}}{b} \]

      3. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(c\right)}{b}} \]

      4. lower-neg.f64 73.6

         \[\leadsto \frac{\color{blue}{-c}}{b} \]

   5. Applied rewrites 73.6%

      \[\leadsto \color{blue}{\frac{-c}{b}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 68.5% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -5e-310) (/ (- b) a) (/ (- c) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e-310) {
		tmp = -b / a;
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-5d-310)) then
        tmp = -b / a
    else
        tmp = -c / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e-310) {
		tmp = -b / a;
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -5e-310:
		tmp = -b / a
	else:
		tmp = -c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -5e-310)
		tmp = Float64(Float64(-b) / a);
	else
		tmp = Float64(Float64(-c) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -5e-310)
		tmp = -b / a;
	else
		tmp = -c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -5e-310], N[((-b) / a), $MachinePrecision], N[((-c) / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -4.999999999999985e-310

   1. Initial program 74.0%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. Add Preprocessing

   3. Taylor expanded in b around -inf

      \[\leadsto \color{blue}{-1 \cdot \frac{b}{a}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{-1 \cdot b}{a}} \]

      2. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{-1 \cdot b}{a}} \]

      3. mul-1-neg N/A

         \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(b\right)}}{a} \]

      4. lower-neg.f64 70.1

         \[\leadsto \frac{\color{blue}{-b}}{a} \]

   5. Applied rewrites 70.1%

      \[\leadsto \color{blue}{\frac{-b}{a}} \]

   if -4.999999999999985e-310 < b

   1. Initial program 30.5%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{-1 \cdot c}{b}} \]

      2. mul-1-neg N/A

         \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(c\right)}}{b} \]

      3. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(c\right)}{b}} \]

      4. lower-neg.f64 73.6

         \[\leadsto \frac{\color{blue}{-c}}{b} \]

   5. Applied rewrites 73.6%

      \[\leadsto \color{blue}{\frac{-c}{b}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 44.1% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (if (<= b -5e-310) (/ (- b) a) 0.0))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e-310) {
		tmp = -b / a;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-5d-310)) then
        tmp = -b / a
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e-310) {
		tmp = -b / a;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -5e-310:
		tmp = -b / a
	else:
		tmp = 0.0
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -5e-310)
		tmp = Float64(Float64(-b) / a);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -5e-310)
		tmp = -b / a;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -5e-310], N[((-b) / a), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if b < -4.999999999999985e-310

   1. Initial program 74.0%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. Add Preprocessing

   3. Taylor expanded in b around -inf

      \[\leadsto \color{blue}{-1 \cdot \frac{b}{a}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{-1 \cdot b}{a}} \]

      2. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{-1 \cdot b}{a}} \]

      3. mul-1-neg N/A

         \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(b\right)}}{a} \]

      4. lower-neg.f64 70.1

         \[\leadsto \frac{\color{blue}{-b}}{a} \]

   5. Applied rewrites 70.1%

      \[\leadsto \color{blue}{\frac{-b}{a}} \]

   if -4.999999999999985e-310 < b

   1. Initial program 30.5%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
   2. Add Preprocessing

   4. Applied rewrites 26.0%

      \[\leadsto \frac{\color{blue}{\frac{\mathsf{fma}\left(-b, b, \mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)\right)}{\sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)} + b}}}{2 \cdot a} \]

   5. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\frac{1}{4} \cdot \frac{-1 \cdot {b}^{2} + {b}^{2}}{a \cdot b}} \]
   6. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{4} \cdot \left(-1 \cdot {b}^{2} + {b}^{2}\right)}{a \cdot b}} \]

      2. distribute-lft1-in N/A

         \[\leadsto \frac{\frac{1}{4} \cdot \color{blue}{\left(\left(-1 + 1\right) \cdot {b}^{2}\right)}}{a \cdot b} \]

      3. metadata-eval N/A

         \[\leadsto \frac{\frac{1}{4} \cdot \left(\color{blue}{0} \cdot {b}^{2}\right)}{a \cdot b} \]

      4. mul0-lft N/A

         \[\leadsto \frac{\frac{1}{4} \cdot \color{blue}{0}}{a \cdot b} \]

      5. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{0}}{a \cdot b} \]

      6. mul0-lft N/A

         \[\leadsto \frac{\color{blue}{0 \cdot {b}^{2}}}{a \cdot b} \]

      7. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{\left(-1 + 1\right)} \cdot {b}^{2}}{a \cdot b} \]

      8. distribute-lft1-in N/A

         \[\leadsto \frac{\color{blue}{-1 \cdot {b}^{2} + {b}^{2}}}{a \cdot b} \]

      9. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{-1 \cdot {b}^{2} + {b}^{2}}{a}}{b}} \]

      10. div-add N/A

          \[\leadsto \frac{\color{blue}{\frac{-1 \cdot {b}^{2}}{a} + \frac{{b}^{2}}{a}}}{b} \]

      11. associate-*r/ N/A

          \[\leadsto \frac{\color{blue}{-1 \cdot \frac{{b}^{2}}{a}} + \frac{{b}^{2}}{a}}{b} \]

      12. distribute-lft1-in N/A

          \[\leadsto \frac{\color{blue}{\left(-1 + 1\right) \cdot \frac{{b}^{2}}{a}}}{b} \]

      13. metadata-eval N/A

          \[\leadsto \frac{\color{blue}{0} \cdot \frac{{b}^{2}}{a}}{b} \]

      14. mul0-lft N/A

          \[\leadsto \frac{\color{blue}{0}}{b} \]

      15. mul0-lft N/A

          \[\leadsto \frac{\color{blue}{0 \cdot \frac{{b}^{2}}{c}}}{b} \]

      16. metadata-eval N/A

          \[\leadsto \frac{\color{blue}{\left(-1 + 1\right)} \cdot \frac{{b}^{2}}{c}}{b} \]

      17. distribute-lft1-in N/A

          \[\leadsto \frac{\color{blue}{-1 \cdot \frac{{b}^{2}}{c} + \frac{{b}^{2}}{c}}}{b} \]

      18. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{-1 \cdot \frac{{b}^{2}}{c} + \frac{{b}^{2}}{c}}{b}} \]

      19. distribute-lft1-in N/A

          \[\leadsto \frac{\color{blue}{\left(-1 + 1\right) \cdot \frac{{b}^{2}}{c}}}{b} \]

      20. metadata-eval N/A

          \[\leadsto \frac{\color{blue}{0} \cdot \frac{{b}^{2}}{c}}{b} \]

      21. mul0-lft 17.7

          \[\leadsto \frac{\color{blue}{0}}{b} \]

   7. Applied rewrites 17.7%

      \[\leadsto \color{blue}{\frac{0}{b}} \]

   8. Taylor expanded in b around 0

      \[\leadsto 0 \]
   9. Step-by-step derivation

   10. Applied rewrites 17.7%

       \[\leadsto 0 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 11.2% accurate, 50.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 0.0)

C

double code(double a, double b, double c) {
	return 0.0;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = 0.0d0
end function

Java

public static double code(double a, double b, double c) {
	return 0.0;
}

Python

def code(a, b, c):
	return 0.0

Julia

function code(a, b, c)
	return 0.0
end

MATLAB

function tmp = code(a, b, c)
	tmp = 0.0;
end

Wolfram

code[a_, b_, c_] := 0.0

Derivation

1. Initial program 49.4%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
2. Add Preprocessing

4. Applied rewrites 24.7%

   \[\leadsto \frac{\color{blue}{\frac{\mathsf{fma}\left(-b, b, \mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)\right)}{\sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)} + b}}}{2 \cdot a} \]

5. Taylor expanded in a around 0

   \[\leadsto \color{blue}{\frac{1}{4} \cdot \frac{-1 \cdot {b}^{2} + {b}^{2}}{a \cdot b}} \]
6. Step-by-step derivation

   1. associate-*r/ N/A

      \[\leadsto \color{blue}{\frac{\frac{1}{4} \cdot \left(-1 \cdot {b}^{2} + {b}^{2}\right)}{a \cdot b}} \]

   2. distribute-lft1-in N/A

      \[\leadsto \frac{\frac{1}{4} \cdot \color{blue}{\left(\left(-1 + 1\right) \cdot {b}^{2}\right)}}{a \cdot b} \]

   3. metadata-eval N/A

      \[\leadsto \frac{\frac{1}{4} \cdot \left(\color{blue}{0} \cdot {b}^{2}\right)}{a \cdot b} \]

   4. mul0-lft N/A

      \[\leadsto \frac{\frac{1}{4} \cdot \color{blue}{0}}{a \cdot b} \]

   5. metadata-eval N/A

      \[\leadsto \frac{\color{blue}{0}}{a \cdot b} \]

   6. mul0-lft N/A

      \[\leadsto \frac{\color{blue}{0 \cdot {b}^{2}}}{a \cdot b} \]

   7. metadata-eval N/A

      \[\leadsto \frac{\color{blue}{\left(-1 + 1\right)} \cdot {b}^{2}}{a \cdot b} \]

   8. distribute-lft1-in N/A

      \[\leadsto \frac{\color{blue}{-1 \cdot {b}^{2} + {b}^{2}}}{a \cdot b} \]

   9. associate-/r* N/A

      \[\leadsto \color{blue}{\frac{\frac{-1 \cdot {b}^{2} + {b}^{2}}{a}}{b}} \]

   10. div-add N/A

       \[\leadsto \frac{\color{blue}{\frac{-1 \cdot {b}^{2}}{a} + \frac{{b}^{2}}{a}}}{b} \]

   11. associate-*r/ N/A

       \[\leadsto \frac{\color{blue}{-1 \cdot \frac{{b}^{2}}{a}} + \frac{{b}^{2}}{a}}{b} \]

   12. distribute-lft1-in N/A

       \[\leadsto \frac{\color{blue}{\left(-1 + 1\right) \cdot \frac{{b}^{2}}{a}}}{b} \]

   13. metadata-eval N/A

       \[\leadsto \frac{\color{blue}{0} \cdot \frac{{b}^{2}}{a}}{b} \]

   14. mul0-lft N/A

       \[\leadsto \frac{\color{blue}{0}}{b} \]

   15. mul0-lft N/A

       \[\leadsto \frac{\color{blue}{0 \cdot \frac{{b}^{2}}{c}}}{b} \]

   16. metadata-eval N/A

       \[\leadsto \frac{\color{blue}{\left(-1 + 1\right)} \cdot \frac{{b}^{2}}{c}}{b} \]

   17. distribute-lft1-in N/A

       \[\leadsto \frac{\color{blue}{-1 \cdot \frac{{b}^{2}}{c} + \frac{{b}^{2}}{c}}}{b} \]

   18. lower-/.f64 N/A

       \[\leadsto \color{blue}{\frac{-1 \cdot \frac{{b}^{2}}{c} + \frac{{b}^{2}}{c}}{b}} \]

   19. distribute-lft1-in N/A

       \[\leadsto \frac{\color{blue}{\left(-1 + 1\right) \cdot \frac{{b}^{2}}{c}}}{b} \]

   20. metadata-eval N/A

       \[\leadsto \frac{\color{blue}{0} \cdot \frac{{b}^{2}}{c}}{b} \]

   21. mul0-lft 11.2

       \[\leadsto \frac{\color{blue}{0}}{b} \]

7. Applied rewrites 11.2%

   \[\leadsto \color{blue}{\frac{0}{b}} \]

8. Taylor expanded in b around 0

   \[\leadsto 0 \]
9. Step-by-step derivation

10. Applied rewrites 11.2%

    \[\leadsto 0 \]
11. Add Preprocessing

Developer Target 1: 99.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left|\frac{b}{2}\right|\\ t_1 := \sqrt{\left|a\right|} \cdot \sqrt{\left|c\right|}\\ t_2 := \begin{array}{l} \mathbf{if}\;\mathsf{copysign}\left(a, c\right) = a:\\ \;\;\;\;\sqrt{t\_0 - t\_1} \cdot \sqrt{t\_0 + t\_1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{hypot}\left(\frac{b}{2}, t\_1\right)\\ \end{array}\\ \mathbf{if}\;b < 0:\\ \;\;\;\;\frac{t\_2 - \frac{b}{2}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{\frac{b}{2} + t\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (fabs (/ b 2.0)))
        (t_1 (* (sqrt (fabs a)) (sqrt (fabs c))))
        (t_2
         (if (== (copysign a c) a)
           (* (sqrt (- t_0 t_1)) (sqrt (+ t_0 t_1)))
           (hypot (/ b 2.0) t_1))))
   (if (< b 0.0) (/ (- t_2 (/ b 2.0)) a) (/ (- c) (+ (/ b 2.0) t_2)))))

C

double code(double a, double b, double c) {
	double t_0 = fabs((b / 2.0));
	double t_1 = sqrt(fabs(a)) * sqrt(fabs(c));
	double tmp;
	if (copysign(a, c) == a) {
		tmp = sqrt((t_0 - t_1)) * sqrt((t_0 + t_1));
	} else {
		tmp = hypot((b / 2.0), t_1);
	}
	double t_2 = tmp;
	double tmp_1;
	if (b < 0.0) {
		tmp_1 = (t_2 - (b / 2.0)) / a;
	} else {
		tmp_1 = -c / ((b / 2.0) + t_2);
	}
	return tmp_1;
}

Java

public static double code(double a, double b, double c) {
	double t_0 = Math.abs((b / 2.0));
	double t_1 = Math.sqrt(Math.abs(a)) * Math.sqrt(Math.abs(c));
	double tmp;
	if (Math.copySign(a, c) == a) {
		tmp = Math.sqrt((t_0 - t_1)) * Math.sqrt((t_0 + t_1));
	} else {
		tmp = Math.hypot((b / 2.0), t_1);
	}
	double t_2 = tmp;
	double tmp_1;
	if (b < 0.0) {
		tmp_1 = (t_2 - (b / 2.0)) / a;
	} else {
		tmp_1 = -c / ((b / 2.0) + t_2);
	}
	return tmp_1;
}

Python

def code(a, b, c):
	t_0 = math.fabs((b / 2.0))
	t_1 = math.sqrt(math.fabs(a)) * math.sqrt(math.fabs(c))
	tmp = 0
	if math.copysign(a, c) == a:
		tmp = math.sqrt((t_0 - t_1)) * math.sqrt((t_0 + t_1))
	else:
		tmp = math.hypot((b / 2.0), t_1)
	t_2 = tmp
	tmp_1 = 0
	if b < 0.0:
		tmp_1 = (t_2 - (b / 2.0)) / a
	else:
		tmp_1 = -c / ((b / 2.0) + t_2)
	return tmp_1

Julia

function code(a, b, c)
	t_0 = abs(Float64(b / 2.0))
	t_1 = Float64(sqrt(abs(a)) * sqrt(abs(c)))
	tmp = 0.0
	if (copysign(a, c) == a)
		tmp = Float64(sqrt(Float64(t_0 - t_1)) * sqrt(Float64(t_0 + t_1)));
	else
		tmp = hypot(Float64(b / 2.0), t_1);
	end
	t_2 = tmp
	tmp_1 = 0.0
	if (b < 0.0)
		tmp_1 = Float64(Float64(t_2 - Float64(b / 2.0)) / a);
	else
		tmp_1 = Float64(Float64(-c) / Float64(Float64(b / 2.0) + t_2));
	end
	return tmp_1
end

MATLAB

function tmp_3 = code(a, b, c)
	t_0 = abs((b / 2.0));
	t_1 = sqrt(abs(a)) * sqrt(abs(c));
	tmp = 0.0;
	if ((sign(c) * abs(a)) == a)
		tmp = sqrt((t_0 - t_1)) * sqrt((t_0 + t_1));
	else
		tmp = hypot((b / 2.0), t_1);
	end
	t_2 = tmp;
	tmp_2 = 0.0;
	if (b < 0.0)
		tmp_2 = (t_2 - (b / 2.0)) / a;
	else
		tmp_2 = -c / ((b / 2.0) + t_2);
	end
	tmp_3 = tmp_2;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Abs[N[(b / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[N[Abs[a], $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[Abs[c], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = If[Equal[N[With[{TMP1 = Abs[a], TMP2 = Sign[c]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], a], N[(N[Sqrt[N[(t$95$0 - t$95$1), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(t$95$0 + t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(b / 2.0), $MachinePrecision] ^ 2 + t$95$1 ^ 2], $MachinePrecision]]}, If[Less[b, 0.0], N[(N[(t$95$2 - N[(b / 2.0), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[((-c) / N[(N[(b / 2.0), $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision]]]]]

Reproduce

herbie shell --seed 2024326 
(FPCore (a b c)
  :name "quadp (p42, positive)"
  :precision binary64
  :herbie-expected 10

  :alt
  (! :herbie-platform default (let ((sqtD (let ((x (* (sqrt (fabs a)) (sqrt (fabs c))))) (if (== (copysign a c) a) (* (sqrt (- (fabs (/ b 2)) x)) (sqrt (+ (fabs (/ b 2)) x))) (hypot (/ b 2) x))))) (if (< b 0) (/ (- sqtD (/ b 2)) a) (/ (- c) (+ (/ b 2) sqtD)))))

  (/ (+ (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)))